A222638 Product, for k <= n, of the squarefree parts of the total number of arrangements of a set with k elements.

1, 2, 10, 10, 650, 211900, 414688300, 56812297100, 6226684574457100, 6142063931090228011000, 60585938964731049213533111000, 1643471023248326636197980531190858000, 12662130715971848810220521992462621415290000, 214329322370515670487612822767624011121300533960940000

Offset: 0

Author

Michel Marcus, Feb 27 2013

Keywords

• nonn

Comments

In their abstract, Luca and Shparlinski write: "In this note, we show that if we write floor(e*n!) = s(n)*u(n)^2, where s(n) is square-free then S(N) = prod(n<=N) has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the m-th power-free part of s(n) as n ranges from 1 to N, where m = 3 is a positive integer. As an application of such results, we give an upper bound on the number of n <= N such that floor(e*n!) is a square."

Links

• F. Luca and I. E. Shparlinski, On the squarefree parts of floor(e*n!), Glasgow Math. J., 49 (2007), 391-403.

Crossrefs

Cf. A000522, A007913, A222637.

Programs

PARI

a(n) = prod(i=1, n, core(i! * polcoef(exp(x + x*O(x^i)) / (1 - x), i)))

Formula

a(n) = Product_{k <= n} A222637(k) = Product_{k <= n} A007913(A000522(k)).